Finite Group Databasebeta

← #6851987

Subgroups — #6851987

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#32155C214yesmaximal, complemented
5#30967C214yesmaximal, complemented
6#29969C212yesmaximal, complemented
7#28816C212yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27822C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic
11#26443C212yesmaximal, complemented
12#26041C212yesmaximal, complemented
13#25819C212yesmaximal, complemented
14#24305C212yesmaximal
15#22629C211yesmaximal, characteristic, complemented
16#22082C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C2×#554—21no—
19C22×#125C2212yes—
20C2×#501C2214yes—
21C2×#481C2211yescharacteristic
22C2×#478C2214yes—
23#2857C2212yes—
24#2857C2211yescharacteristic
25#2564—22no—
26#2556—22no—
27#2552—21no—
28#2552—21no—
29#2552—22no—
30#2552C2212yes—
31#2552C2212yes—
32#2552C2214yes—
33#2551C2211yescharacteristic
34#2542C2211yescharacteristic
35#2506C2212yes—
36#2492—21no—
37#2452C2212yes—
38#2417—21no—
39C2×#398—22no—
40#2365C2211yescharacteristic
41#2360—21no—
42D4×C4⋊C4C2211yescharacteristic
43C4×#121C2211yescharacteristic
44C4×#121C2212yes—
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384—21no—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2212yes—
53#2093C2212yes—
54#2087C2214yes—
55#2087C2214yes—
56#2083C2214yes—
57#2059C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2057C2214yes—
61#2025C2214yes—
62#1992C2214yes—
63#1990C2212yes—
64#1989C2214yes—
65#1988C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1944C2214yes—
69#1939C2212yes—
70#1936C2214yes—
71#1926C2212yes—
72#1919C2214yes—
73#1919C2212yes—
74#1919C2214yes—
75#1917C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1862C2214yes—
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1845C2212yes—
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#125—48no—
103C2×#125C2314yes—
104C2×#125C2314yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×#121C2311yescharacteristic
111C2×C4×Q8—21no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#501—48no—
131#481Q818yescomplemented
132#478—48no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#399—22no—
151#398—22no—
152#398—24no—
153#398—24no—
154#398—24no—
155#398—24no—
156#396—22no—
157#395—22no—
158#393—22no—
159#391—21no—
160#391—21no—
161#391—22no—
162#390C2311yescharacteristic
163#390C2312yes—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2311yescharacteristic
179#388—22no—
180#388—21no—
181#387—22no—
182#387—22no—
183#386—21no—
184#386—22no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384—24no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#380—21no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253#125—28no—
254#125—28no—
255#125—44no—
256#125—44no—
257#125—48no—
258#125—48no—
259#125—48no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—48no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265#121—22no—
266#121C2×Q814yes—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—44no—
270C4×D4—48no—
271C4×D4—44no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×Q814yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—44no—
347#107—44no—
348#107—44no—
349#106—28no—
350#106—24no—
351#106C2×Q818yes—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#102—24no—
355#102—44no—
356#102C2×Q818yes—
357C22⋊C8—22no—
358C22⋊C8—24no—
359C22⋊C8C2×Q814yes—
360#97—44no—
361C4×C8—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—21no—
380C24C22×D411yescharacteristic
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C22×Q812yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4AGL(1,ℤ/8)14yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×Q812yes—
441C22×C42−1+412yes—
442C22×C4C2511yescharacteristic
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8C22×Q814yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—82no—
491C4⋊C4C22×Q814yes—
492C4⋊C4C22×Q812yes—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—24no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—81no—
502C22⋊C4—82no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—42no—
509C42—42no—
510C42—44no—
511C42—44no—
512C42—42no—
513C42—44no—
514C422−1+414yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23D4×Q812yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23#57812yes—
529C23C23×D411yescharacteristic
530Q8—84no—
531Q8—164no—
532D4—24no—
533D4—48no—
534D4—42no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—84no—
590C2×C4—164no—
591C2×C4#53612yes—
592C2×C4#53612yes—
593C2×C4D4×Q812yes—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4#57512yes—
597C2×C4C23×Q811yescharacteristic
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4C2×2−1+412yes—
600C2×C4C2×2−1+411yescharacteristic
601C8—24no—
602C8—44no—
603C8—42no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#338614yes—
618C22C2×#53611yescharacteristic
619C22C2×D4×Q811yescharacteristic
620C22#348411yescharacteristic
621C22C22×AGL(1,ℤ/8)11yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57811yescharacteristic
624C22#358511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—41no—
636C4—84no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4#323414yes—
642C4#350312yes—
643C4#355212yes—
644C2—24no—
645C2—82no—
646C2#3959211yescharacteristic
647C2C2×#323411yescharacteristic
648C2#6087411yescharacteristic
649C2C2×#338611yescharacteristic
650C2#6136711yescharacteristic
651C2#6158711yescharacteristic
652C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33147C2semidirect
4. #32155C2semidirect
5. #30967C2semidirect
6. #29969C2semidirect
7. #28816C2semidirect
8. #27950C2semidirect
9. #27822C2semidirect
11. #26443C2semidirect
12. #26041C2semidirect
13. #25819C2semidirect
15. #22629C2semidirect
131. #481Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365612
C2×#25522
#331472
#321552
#309672
#299692
#288162
#279502
#278222
#276772
#264432
#260412
#258192
#243052
#226292
#220822
C4×#4812